Problem: Rewrite as a simplified fraction. ${1.3\overline{7} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}100x &= 137.7777...\\ 10x &= 13.7777...\end{align*}}$ Subtracting the two equations, we have: ${90x = 124}$ ${x = \dfrac{124}{90}} $ ${x = \dfrac{62}{45}} \text{or} {1 \dfrac{17}{45}}$